Pass over S2, S3; Ended up saving a column or so
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@ -74,6 +74,7 @@ Since $\semNX$-PDBs $\pxdb$ are a complete representation system for $\semN$-PDB
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\subsection{Proof of~\Cref{prop:expection-of-polynom}}
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\label{subsec:expectation-of-polynom-proof}
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\BG{TODO}
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\subsection{Supplementary Material for~\Cref{def:tidbs-and-bidbs}}\label{subsec:supp-mat-ti-bi-def}
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@ -291,7 +291,7 @@ Our hardness results follow by considering a suitable generalization of the line
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The starting point of our approximation algorithm was the simple observation that for any lineage polynomial $\poly(X_1,\dots,X_n)$, we have $\rpoly(1,\dots,1)=Q(1,\dots,1)$ and if all the coefficients of $\poly$ are constants, then $\poly(X_1,\dots,X_n)$ (which can be easily computed in linear time) is a $p^k$ approximation to the value $\rpoly(p,\dots,p)$ that we are after. If $p$ and $k=\deg(\poly)$ are constants, then this gives a constant factor approximation. We then use sampling to get a better approximation factor of $(1\pm \eps)$: we sample monomials from $\poly(X_1,\dots,X_n)$ and do an appropriate weighted sum of their coefficients. Standard tail bounds then allow us to get our desired approximation scheme. To get a linear runtime, it turns out that we need the following properties from our compressed representation of $\poly$: (i) be able to compute $\poly(X_1,\dots,X_n)$ in linear time and (ii) be able to sample monomials from $\poly(X_1,\dots,X_n)$ quickly as well. For the ease of exposition, we start off with expression trees (see~\Cref{fig:intro-q2-etree} for an example) and show that they satisfy both of these properties. Later we show that it is easy to show that these properties also extend to polynomial circuits as well (we essentially show that in the required time bound, we can simulate access to the `unrolled' expression tree by considering the polynomial circuit).
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We also formalize our claim that, since our approximation algorithm runs in time linear in the size of the polynomial circuit, we can approximate the expected output tuple multiplicities with only a $O(\log{Z})$ overhead (where $Z$ is the number of output tuples) over the runtime of a broad class of query processing algorithms. We also observe that our results trivially extend to the problem of computing higher moments of the tuple multiplicity (instead of just the expectation).
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We also formalize our claim that, since our approximation algorithm runs in time linear in the size of the polynomial circuit, we can approximate the expected output tuple multiplicities with only a $O(\log{Z})$ overhead (where $Z$ is the number of output tuples) over the runtime of a broad class of query processing algorithms. We also observe that our results trivially extend to higher moments of the tuple multiplicity (instead of just the expectation).
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\paragraph{Paper Organization.} We present some relevant background and setup our notation in~\Cref{sec:background}. We present our hardness results in~\Cref{sec:hard} and our approximation algorithm in~\Cref{sec:algo}. We present some (easy) generalizations of our results in~\Cref{sec:gen}. We do a quick overview of related work in~\Cref{sec:related-work} and conclude with some open questions in~\Cref{sec:concl-future-work}.
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@ -147,6 +147,7 @@
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%using \wVec for world bit vector notation
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\newcommand{\poly}{Q}
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\newcommand{\rpoly}{\widetilde{Q}}%r for reduced as in reduced 'Q'
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\newcommand{\polyForTuple}{\poly_{\tup}}
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\newcommand{\out}{output}%output aggregation over the output vector
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\newcommand{\numocc}[2]{\#\left(#1, #2\right)}
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@ -1,10 +1,10 @@
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%root:main.tex
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%!TEX root=./main.tex
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\section{Hardness of exact computation}
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\label{sec:hard}
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\AH{The notation used here is different than in~\Cref{sec:background}, in particular~\Cref{eq:expect-q-nx}. Maybe we should decide on a notation and try to stick to it as much as possible?}
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\BG{We sometimes use $\expct_{\vct{X} \sim P}$ sometimes $\expct_{\vct{X}}$}
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In this section, we will prove that computing $\expct\limits_{\vct{X} \sim \pd}\pbox{\poly(\vct{X})}$ for a \tis-lineage polynomial $\poly(\vct{X})$ generated from a project-join query is \sharpwonehard. Note that this implies hardness for \bis and general $\semNX$-PDBs. Furthermore, we demonstrate \Cref{sec:single-p} that the problem remains hard, even if $\pd(X_i) = p$ for all $X_i$ and some fixed valued $p$ as long as these conjectures hold. Finally, using popular hardness conjectures in fine-grained complexity we show that if these conjectures hold and except for the trivial choices of $p \in \{0,1\}$, the problem is hard for any given $p$.
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In this section, we will prove that computing $\expct\limits_{\vct{X} \sim \pd}\pbox{\poly(\vct{X})}$ for a \ti-lineage polynomial $\poly(\vct{X})$ generated from a project-join query is \sharpwonehard. Note that this implies hardness for \bis and general $\semNX$-PDBs. Furthermore, we demonstrate \Cref{sec:single-p} that the problem remains hard, even if $\pd(X_i) = p$ for all $X_i$ and some fixed valued $p$ as long as these conjectures hold. Finally, using popular hardness conjectures in fine-grained complexity we show that if these conjectures hold and except for the trivial choices of $p \in \{0,1\}$, the problem is hard for any given $p$.
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% We would like to argue for a compressed version of $\poly(\vct{X})$, in general $\expct\limits_{\vct{X} \sim \pd}\pbox{\poly(\vct{X})}$ even for tis, cannot be computed in linear time. We will argue two flavors of such a hardness result. In Section~\ref{sec:multiple-p}, we argue that computing the expected value exactly for all query polynommials $\poly(\vct{X})$ for multiple values of $p$ is \sharpwonehard. However, this does not rule out the possibility of being able to solve the problem for a any {\em fixed} value of $p$ in linear time. In Section~\ref{sec:single-p}, we rule out even this possibility (based on some popular hardness conjectures in fine-grained complexity).
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@ -12,10 +12,9 @@ In this section, we will prove that computing $\expct\limits_{\vct{X} \sim \pd}\
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\subsection{Preliminaries}
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Our hardness results are based on (exactly) counting the number of occurrences of a fixed graph $H$ as a subgraph in $G$. Let $\numocc{G}{H}$ denote the number of occurrences of pattern $H$ in graph $G$. %, where, for example, $\numocc{G}{\ed}$ means the number of single edges in $G$.
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In particular, we will consider the problems of computing the following counts (given $G$ as an input in its adjacency list representation): $\numocc{G}{\tri}$ (the number of triangles), $\numocc{G}{\threepath}$ (the number of $3$-paths), $\numocc{G}{\threedis}$ (the number of $3$-matchings or collection of three node disjoint edges) and its generalization $\numocc{G}{\kmatch}$ (the number of $k$-matchings or collections of $k$ node-disjoint edges).
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Our hardness result in \Cref{sec:multiple-p} is based on the following hardness result:
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In particular, we will consider the problems of computing the following counts (given $G$ as an input in its adjacency list representation): $\numocc{G}{\tri}$ (the number of triangles), $\numocc{G}{\threepath}$ (the number of $3$-paths), $\numocc{G}{\threedis}$ (the number of $3$-matchings or collection of three node-disjoint edges) and its generalization $\numocc{G}{\kmatch}$ (the number of $k$-matchings or collections of $k$ node-disjoint edges).
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%
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Our hardness result in \Cref{sec:multiple-p} is based on the following result:
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Theorem}[\cite{k-match}]
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@ -26,7 +25,7 @@ Given a positive integer $k$ and an undirected graph $G$ with no self-loops or
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The above result means that we cannot hope to count the number of $k$-matchings in $G=(V,E)$ in time $f(k)\cdot |V|^{O(1)}$ for any function $f$. In fact, all known algorithms to solve this problem take time $|V|^{\Omega(k)}$.
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%
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Our hardness result in Section~\ref{sec:single-p} is based on the following conjectured hardness result:
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -39,24 +38,24 @@ There exists a constant $\eps_0>0$ such that given an undirected graph $G=(V,E)$
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Based on the so called {\em Triangle detection hypothesis} (cf.~\cite{triang-hard}), which states that detection whether $G$ has a triangle or not takes time $\Omega\inparen{|E|^{4/3}}$, implies that in Conjecture~\ref{conj:graph} we can take $\eps_0\ge \frac 13$.
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\AR{Need to add something about 3-paths and 3-matchings as well.}
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Both of our hardness results use a query polynomial that is based on a simple encoding of the edges of a graph.
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To prove our hardness result, consider a graph $G(V, E)$, where $|E| = m$, $|V| = \numvar$. Our query polynomial will have a variable $X_i$ for every $i$ in $[\numvar]$.
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Now consider the polynomial
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Both of our hardness results rely on a simple query polynomial encoding of the edges of a graph.
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To prove our hardness result, consider a graph $G(V, E)$, where $|E| = m$, $|V| = \numvar$. Our query polynomial has a variable $X_i$ for every $i$ in $[\numvar]$.
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Consider the polynomial
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\[\poly_{G}(\vct{X}) = \sum\limits_{(i, j) \in E} X_i \cdot X_j.\]
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The hard polynomial for our problem will be a suitable power $k\ge 3$ of the polynomial above, i.e.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Definition}
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Let $G=([n],E)$ be a graph. Then for any $\kElem\ge 1$, define
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For any graph $G=([n],E)$ and $\kElem\ge 1$, define
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\[\poly_{G}^\kElem(X_1,\dots,X_n) = \left(\sum\limits_{(i, j) \in E} X_i \cdot X_j\right)^\kElem.\]
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\end{Definition}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Our hardness results only need a \ti instance and further, we consider the special case when all the tuple probabilities (probabilities assigned by to $X_i$ by $\vct{p}$) are the same value. It is not too hard to see that we can encode the above polynomial in an expression tree of size $\Theta(km)$.
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Our hardness results only need a \ti instance; We also consider the special case when all the tuple probabilities (probabilities assigned by to $X_i$ by $\vct{p}$) are the same value. Note that this polynomial can be encoded in an expression tree of size $\Theta(km)$.
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Following up on the discussion around Example~\ref{ex:intro}, it is easy to see that $\poly_{G}^\kElem(\vct{X})$ is the query polynomial corresponding to the following query:
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Following on Example~\ref{ex:intro}, it is easy to see that $\poly_{G}^\kElem(\vct{X})$ is the query polynomial corresponding to the query:
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\[\poly^k_G:- R(A_1),E(A_1,B_1),R(B_1),\dots,R(A_\kElem),E(A_\kElem,B_\kElem),R(B_\kElem)\]
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where generalizaing the PDB instance in Example~\ref{ex:intro}, relation $R$ has $n$ tuples corresponding to each vertex in $V=[n]$ each with probability $p$ and $E(A,B)$ has tuples corresponding to the edges in $E$ (each with probability of $1$).\footnote{Technically, $\poly_{G}^\kElem(\vct{X})$ should have variables corresponding to tuples in $E$ as well but since they always are present with probability $1$, we drop those. Our argument also works when all the tuples in $E$ also are present with probability $p$ but to simplify notation we assign probability $1$ to edges.}
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where generalizaing the PDB instance in Example~\ref{ex:intro}, relation $R$ has $n$ tuples corresponding to each vertex in $V=[n]$ each with probability $p$ and $E(A,B)$ has tuples corresponding to the edges in $E$ (each with probability of $1$).\footnote{Technically, $\poly_{G}^\kElem(\vct{X})$ should have variables corresponding to tuples in $E$ as well, but since they always are present with probability $1$, we drop those. Our argument also works when all the tuples in $E$ also are present with probability $p$ but to simplify notation we assign probability $1$ to edges.}
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Note that this imples that our hard query polynomial can be created from a project-join query -- by contrast our approximation algorithm in \Cref{sec:algo} can handle lineage polynomials generated by union of select-project-join queries. % (i.e. we do not need union or select operator to derive our hardness result).
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Note that this imples that our hard query polynomial can be created from a project-join query -- by contrast our approximation algorithm in \Cref{sec:algo} can handle lineage polynomials generated by union of select-project-join (SPJU) queries. % (i.e. we do not need union or select operator to derive our hardness result).
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%\AR{need discussion on the `tightness' of various params. First, this is for degree 6 poly-- while things are easy for say deg 2. Second this is for any fixed p. Finally, we only need project-join queries to get the hardness results. Also need to compare this with the generality of the approx upper bound results.}
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@ -71,7 +70,7 @@ Computing $\rpoly_G^\kElem(\prob_i,\dots,\prob_i)$ for arbitrary $G$ and any $(2
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\end{Theorem}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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We will prove the above result by reduction from the problem of computing the number of $k$-matchings in $G$. Given the current best-known algorithm for this counting problem, our results imply that unless the state-of-the-art $k$-matching algorithms are improved, we cannot hope to solve our problem in time better than $\Omega_k\inparen{m^{k/2}}$, which is only quadratically faster than expanding $\poly_{G}^\kElem(\vct{X})$ into its \abbrSMB form and then using \Cref{cor:expct-sop}. By contrast the approximation algorithm we present in \Cref{sec:algo} runtime is in$O_k\inparen{m}$ for this query (since it runs in linear-time on all lineage polynomials).
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We will prove the above result by reduction from the problem of computing the number of $k$-matchings in $G$. Given the current best-known algorithm for this counting problem, our results imply that unless the state-of-the-art $k$-matching algorithms are improved, we cannot hope to solve our problem in time better than $\Omega_k\inparen{m^{k/2}}$, which is only quadratically faster than expanding $\poly_{G}^\kElem(\vct{X})$ into its \abbrSMB form and then using \Cref{cor:expct-sop}. By contrast the approximation algorithm we present in \Cref{sec:algo} has runtime $O_k\inparen{m}$ for this query (since it runs in linear-time on all lineage polynomials).
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As mentioned earlier, we prove our hardness result by presenting a reduction from the problem of couting $\kElem$-matchings in a graph:
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\begin{Lemma}\label{lem:qEk-multi-p}
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@ -92,24 +91,30 @@ We first argue that $\rpoly_{G}^\kElem(\prob,\ldots, \prob) = \sum\limits_{i = 0
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%\sum_{\substack{(i_1, j_1),\\\cdots,\\(i_\kElem, j_\kElem) \in E}}X_{i_1}X_{j_1}\cdots X_{i_\kElem}X_{j_\kElem}
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%\end{equation*}
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%Since each of $(i_1, j_1),\ldots, (i_\kElem, j_\kElem)$ are from $E$, it follows that the set of $\kElem!$ permutations of the $\kElem$ $X_iX_j$ pairs which form the monomial products are of degree $2\kElem$ with the number of distinct variables in an arbitrary monomial $\leq 2\kElem$.
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By definition, $\rpoly_{G}^{\kElem}(\vct{X})$ sets every exponent $e > 1$ to $e = 1$, which means that $\degree(\rpoly_{G}^\kElem)\le \degree(\poly_G^\kElem)=2k$. Thus, if we think of $\prob$ as a variable, then $\rpoly_{G}^{\kElem}(\prob,\dots,\prob)$ is a univariate polynomial of degree at most $\degree(\rpoly_{G}^\kElem)\le 2k$. Thus, we can write
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By definition, $\rpoly_{G}^{\kElem}(\vct{X})$ sets every exponent $e > 1$ to $e = 1$, which means that $\degree(\rpoly_{G}^\kElem)\le \degree(\poly_G^\kElem)= 2k$. Thus, if we think of $\prob$ as a variable, then $\rpoly_{G}^{\kElem}(\prob,\dots,\prob)$ is a univariate polynomial of degree at most $\degree(\rpoly_{G}^\kElem)\le 2k$. Thus, we can write
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%thereby shrinking the degree a monomial product term in the SOP form of $\poly_{G}^{\kElem}(\vct{X})$ to the exact number of distinct variables the monomial contains. This implies that $\rpoly_{G}^\kElem$ is a polynomial of degree $2\kElem$ and hence $\rpoly_{G}^\kElem(\prob,\ldots, \prob)$ is a polynomial in $\prob$ of degree $2\kElem$. Then it is the case that
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\begin{equation*}
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\rpoly_{G}^{\kElem}(\prob,\ldots, \prob) = \sum_{i = 0}^{2\kElem} c_i \prob^i
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\end{equation*}
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We note that $c_i$ is {\em exactly} the number of monomials in the SOP\BG{\abbrSMB?} expansion of $\poly_{G}^{\kElem}(\vct{X})$ composed of $i$ distinct variables.%, with $\prob$ substituted for each distinct variable
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We note that $c_i$ is {\em exactly} the number of monomials in the SMB %\BG{\abbrSMB?}
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expansion of $\poly_{G}^{\kElem}(\vct{X})$ composed of $i$ distinct variables.%, with $\prob$ substituted for each distinct variable
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\footnote{Since $\rpoly_G^\kElem(\vct{X})$ does not have any monomial with degree $< 2$, it is the case that $c_0 = c_1 = 0$ but for the sake of simplcity we will ignore this observation.}
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Given that we then have $2\kElem + 1$ distinct values of $\rpoly_{G}^\kElem(\prob,\ldots, \prob)$ for $0\leq i\leq2\kElem$, it follows that
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%we then have $2\kElem + 1$ distinct rows of the form $\prob_i^0\ldots\prob_i^{2\kElem}$ which form a matrix $M$.
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we have a linear system of the form $\vec{M} \cdot \vct{c} = \vct{b}$ where the $i$th row of $\vec{M}$ is $\inparen{\prob_i^0\ldots\prob_i^{2\kElem}}$, $\vct{c}$ is the coefficient vector $\inparen{c_0,\ldots, c_{2\kElem}}$, and $\vct{b}$ is the vector such that $\vct{b}[i] = \rpoly_{G}^\kElem(\prob_i,\ldots, \prob_i)$. In other words, matrix $\vec{M}$ is the Vandermonde matrix, from which it follows that we have a matrix with full rank (since the $p_i$'s are distinct), and we can solve the linear system in $O(k^3)$ time (say using Gaussian Elimination) to determine $\vct{c}$ exactly. Thus, after $O(k^3)$ work, we know $\vct{c}$ and in particular, $c_{2k}$ exactly. Next we show why we can compute $\numocc{G}{\kmatch}$ from $c_{2k}$ in $O(1)$ additional time.
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we have a linear system of the form $\vec{M} \cdot \vct{c} = \vct{b}$ where the $i$th row of $\vec{M}$ is $\inparen{\prob_i^0\ldots\prob_i^{2\kElem}}$, $\vct{c}$ is the coefficient vector $\inparen{c_0,\ldots, c_{2\kElem}}$, and $\vct{b}$ is the vector such that $\vct{b}[i] = \rpoly_{G}^\kElem(\prob_i,\ldots, \prob_i)$.
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In other words, matrix $\vec{M}$ is the Vandermonde matrix, from which it follows that we have a matrix with full rank (the $p_i$'s are distinct), and we can solve the linear system in $O(k^3)$ time (e.g., using Gaussian Elimination) to determine $\vct{c}$ exactly.
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Thus, after $O(k^3)$ work, we know $\vct{c}$ and in particular, $c_{2k}$ exactly.
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Next, we show why we can compute $\numocc{G}{\kmatch}$ from $c_{2k}$ in $O(1)$ additional time.
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%Denote the number of $\kElem$-matchings in $G$ as $\numocc{G}{\kmatch}$.
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We claim that $c_{2\kElem}$ is $\kElem! \cdot \numocc{G}{\kmatch}$. This can be seen intuitively by looking at the original factorized representation
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\[\poly_{G}^\kElem(\vct{X}) = \sum_{\substack{(i_1, j_1),\\\cdots,\\(i_\kElem, j_\kElem) \in E}}X_{i_1}X_{j_1}\cdots X_{i_\kElem}X_{j_\kElem},\]
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where across each of the $\kElem$ products, an arbitrary $\kElem$-matching can be selected $\prod_{i = 1}^\kElem \kElem = \kElem!$ times. Next, note that each $\kElem$-matching $(i_1, j_1)\ldots$ $(i_k, j_k)$ in $G$ corresponds to the monomial $\prod_{\ell = 1}^\kElem X_{i_\ell}X_{j_\ell}$ in $\poly_{G}^\kElem(\vct{X})$, with all indexes distinct. %Since each index is distinct, then each variable has an exponent $e = 1$ and this monomial survives in $\rpoly_{G}^{\kElem}(\vct{X})$ Since $\rpoly$ contains only exponents $e \leq 1$, the only degree $2\kElem$ terms that can exist in $\rpoly_{G}^\kElem$ are $\kElem$-matchings since every other monomial in $\poly_{G}^\kElem(\vct{X})$ has strictly less than $2\kElem$ distinct variables, which, as stated earlier implies that every other non-$\kElem$-matching monomial in $\rpoly_{G}^\kElem(\vct{X})$ has degree $< 2\kElem$.
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Second, the only surviving monomials $\prod_{\ell = 1}^\kElem X_{i_\ell}X_{j_\ell}$ of degree exactly $2k$ in $\rpoly_{G}^{\kElem}(\vct{X})$ must have that all of $i_1,j_1,\dots,i_\kElem,j_\kElem$ are distinct in $\poly_{G}^{\kElem}(\vct{X})$. Then, by the last two statements, only monomials composed of $2k$ distinct variables in $\poly_{G}^{\kElem}(\vct{X})$ (and hence of degree $2\kElem$ in $\rpoly_{G}^{\kElem}(\vct{X})$) correspond to a $k$-matching in $G$.
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\[\poly_{G}^\kElem(\vct{X}) = \sum_{\substack{(i_1, j_1),\cdots,(i_\kElem, j_\kElem) \in E}}X_{i_1}X_{j_1}\cdots X_{i_\kElem}X_{j_\kElem},\]
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where across each of the $\kElem$ products, an arbitrary $\kElem$-matching can be selected $\prod_{i = 1}^\kElem i = \kElem!$ times.
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Next, note that each $\kElem$-matching $(i_1, j_1)\ldots$ $(i_k, j_k)$ in $G$ corresponds to the monomial $\prod_{\ell = 1}^\kElem X_{i_\ell}X_{j_\ell}$ in $\poly_{G}^\kElem(\vct{X})$, with distinct indexes. %Since each index is distinct, then each variable has an exponent $e = 1$ and this monomial survives in $\rpoly_{G}^{\kElem}(\vct{X})$ Since $\rpoly$ contains only exponents $e \leq 1$, the only degree $2\kElem$ terms that can exist in $\rpoly_{G}^\kElem$ are $\kElem$-matchings since every other monomial in $\poly_{G}^\kElem(\vct{X})$ has strictly less than $2\kElem$ distinct variables, which, as stated earlier implies that every other non-$\kElem$-matching monomial in $\rpoly_{G}^\kElem(\vct{X})$ has degree $< 2\kElem$.
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Second, the only surviving monomials $\prod_{\ell = 1}^\kElem X_{i_\ell}X_{j_\ell}$ of degree exactly $2k$ in $\rpoly_{G}^{\kElem}(\vct{X})$ must have that all of $i_1,j_1,\dots,i_\kElem,j_\kElem$ are distinct in $\poly_{G}^{\kElem}(\vct{X})$.
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By the last two statements, only monomials composed of $2k$ distinct variables in $\poly_{G}^{\kElem}(\vct{X})$ (and hence of degree $2\kElem$ in $\rpoly_{G}^{\kElem}(\vct{X})$) correspond to a $k$-matching in $G$.
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%It has already been established above that a $\kElem$-matching ($\kmatch$) has coefficient $c_{2\kElem}$. As noted, a $\kElem$-matching occurs when there are $\kElem$ edges, $e_1, e_2,\ldots, e_\kElem$, such that all of them are disjoint, i.e., $e_1 \neq e_2 \neq \cdots \neq e_\kElem$. In all $\kElem$ factors of $\poly_{G}^\kElem(\vct{X})$ there are $k$ choices from the first factor to select an edge for a given $\kElem$ matching, $\kElem - 1$ choices in the second factor, and so on throughout all the factors, yielding $\kElem!$ duplicate terms for each $\kElem$ matching in the expansion of $\poly_{G}^\kElem(\vct{X})$.
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Notice that %we have $\kElem!$ duplicates of
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@ -3,9 +3,8 @@
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%\onecolumn
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\subsection{Reduced Polynomials and Equivalences}
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Since we have shown that computing the expected multiplicity of a query result tuple is equivalent to computing the expectation of a polynomial (for that tuple) given a probability distribution over all possible assignments of variables in the polynomial to $\{0,1\}$, we focus on this problem exclusively from now on.
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We now introduce some basic terminology for polynomials and then develop a reduced normal form for polynomials that preserves a polynomial expectation for probability distributions that stem from \bis or \tis.
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Let us use the expression $(X + Y)^2$ as a running example in this section.
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We now introduce some terminology for polynomials and develop a reduced form for polynomials --- a closed form of the polynomial's expectation over probability distributions derived from a \bi or \ti.
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Throughout, we will use $(X + Y)^2$ as a running example.
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% \begin{Definition}[Monomial]\label{def:monomial}
|
||||
|
|
@ -18,42 +17,48 @@ Let us use the expression $(X + Y)^2$ as a running example in this section.
|
|||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{Definition}[Standard Monomial Basis]\label{def:smb}
|
||||
A monomial is a product of a set of variables, each raised to a non-negative integer power.
|
||||
A polynomial is in \termSMB (\abbrSMB) when it is of the form:
|
||||
A monomial is a product of variable terms, each raised to a non-negative integer power.
|
||||
A polynomial in \termSMB (\abbrSMB) has the form:
|
||||
\[
|
||||
\sum_{i=1}^n c_i \cdot m_i
|
||||
\]
|
||||
where each $c_i$ is a positive integer and each $m_i$ is a monomial and $m_i \neq m_j$ for $i \neq j$. Given a polynomial $\poly$ we denote its \abbrSMB as $\smbOf{\poly}$.
|
||||
where each $c_i$ is a positive integer and each $m_i$ is a monomial and $m_i \neq m_j$ for $i \neq j$. The \abbrSMB of a polynomial $\poly$ is $\smbOf{\poly}$.
|
||||
% fully expanded out such that no product of sums exist and where each unique monomial appears exactly once.
|
||||
\end{Definition}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
The \abbrSMB for the running example is $X^2 +2XY + Y^2$. While $X^2 + XY + XY + Y^2$ is an expanded form of the expression, it is not the standard monomial basis since $XY$ appears more than once.
|
||||
|
||||
\BG{Maybe inline degree?}
|
||||
% \BG{Maybe inline degree?}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{Definition}[Degree]\label{def:degree}
|
||||
The degree of polynomial $\poly(\vct{X})$ is the maximum sum of the exponents of a monomial, over all monomials in $\smbOf{\poly(\vct{X})}$.
|
||||
The degree of polynomial $\poly(\vct{X})$ is the maximum sum of exponents, over all monomials in $\smbOf{\poly(\vct{X})}$.
|
||||
\end{Definition}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
The degree of the running example polynomial is $2$. In this paper we consider only finite degree polynomials.
|
||||
|
||||
%
|
||||
% Throughout this paper, we also make the following \textit{assumption}.
|
||||
|
||||
%
|
||||
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% \begin{Assumption}\label{assump:poly-smb}
|
||||
% All polynomials considered are in standard monomial basis, i.e., $\poly(\vct{X}) = \sum\limits_{\vct{d} \in \mathbb{N}^\numvar}q_d \cdot \prod\limits_{i = 1, d_i \geq 1}^{\numvar}X_i^{d_i}$, where $q_d$ is the coefficient for the monomial encoded in $\vct{d}$ and $d_i$ is the $i^{th}$ element of $\vct{d}$.
|
||||
% \end{Assumption}
|
||||
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
We call a polynomial $\query(\vct{X})$ a \emph{\bi-lineage polynomial} (\emph{\ti-lineage polynomial}, or simply lineage polynomial), if
|
||||
\AH{Why is it required for the tuple to be n-ary? I think this slightly confuses me since we have n tuples.} there exists an n-ary $\raPlus$ query $\query$, \bi $\pxdb$ (\ti $\pxdb$, or $\semNX$-PDB $\pxdb$), and n-ary tuple $\tup$ such that $\query(\vct{X}) = \query(\pxdb)(\tup)$. % Before proceeding, note that the following is assume that polynomials are \bis (which subsume \tis as a special case).
|
||||
Note the \tis are a special case of \bis and, thus, the following applies to \tis as well.
|
||||
Recall that in a \bi $\pxdb$ with tuples $t_1, \ldots, t_n$, each input tuple $t_i$ is annotated with a unique variable $X_i$. The tuples of $\pxdb$ are partitioned into $\ell$ blocks $\block_1, \ldots, \block_\ell$ and each tuple $t_i$ is associated with a probability $\prob(\tup_i) = \pd[X_i = 1]$. Together with the assumption that blocks are assumed to be independent and tuples from the same block are disjoint events, $\prob$ and the blocks induce the probability distribution $\pd$ of $\pxdb$.
|
||||
%
|
||||
We call a polynomial $\query(\vct{X})$ a \emph{\bi-lineage polynomial} (resp., \emph{\ti-lineage polynomial}, or simply lineage polynomial), if
|
||||
%\AH{Why is it required for the tuple to be n-ary? I think this slightly confuses me since we have n tuples.}
|
||||
% OK: agreed w/ AH, this can be treated as implicit
|
||||
there exists a $\raPlus$ query $\query$, \bi $\pxdb$ (\ti $\pxdb$, or $\semNX$-PDB $\pxdb$), and tuple $\tup$ such that $\query(\vct{X}) = \query(\pxdb)(\tup)$. % Before proceeding, note that the following is assume that polynomials are \bis (which subsume \tis as a special case).
|
||||
As they are a special case of \bis, the following applies to \tis as well.
|
||||
Recall that in a \bi $\pxdb$ with tuples $t_1, \ldots, t_n$, each input tuple $t_i$ is annotated with a unique variable $X_i$.
|
||||
Tuples of $\pxdb$ are partitioned into $\ell$ blocks $\block_1, \ldots, \block_\ell$ where tuple $t_i$ is associated with a probability $\prob(\tup_i) = \pd[X_i = 1]$.\footnote{
|
||||
Note the deviation from the more common approach of defining a single independent, $[\abs{\block_i}+1]$-valued variable per block; Here we define $\abs{\block_i}$ correlated variables per block.
|
||||
}
|
||||
Because blocks are independent and tuples from the same block are disjoint, $\prob$ and the blocks induce the probability distribution $\pd$ of $\pxdb$.
|
||||
We will write a \bi-lineage polynomial $\poly(\vct{X})$ for a \bi with $\ell$ blocks as
|
||||
$\poly(\vct{X})$ = $\poly(X_{\block_1, 1},\ldots, X_{\block_1, \abs{\block_1}},$ $\ldots, X_{\block_\ell, \abs{\block_\ell}})$, where $\abs{\block_i}$ denotes the size of $\block_i$, and $\block_{i, j}$ denotes tuple $j$ residing in block $i$ for $j$ in $[\abs{\block_i}]$.
|
||||
\SF{Where is $\block_{i, j}$ used? Is it $X_{\block_{1, 1}}$ or $X_{\block_1, 1}$ ?}
|
||||
$\poly(\vct{X})$ = $\poly(X_{\block_1, 1},\ldots, X_{\block_1, \abs{\block_1}},$ $\ldots, X_{\block_\ell, \abs{\block_\ell}})$, where $\abs{\block_i}$ denotes the size of $\block_i$, and $X_{i, j}$ denotes the annotation of tuple $j$ residing in block $i$ for $j$ in $[\abs{\block_i}]$.
|
||||
%\SF{Where is $\block_{i, j}$ used? Is it $X_{\block_{1, 1}}$ or $X_{\block_1, 1}$ ?}
|
||||
% and the probability distribution of $\pxdb$ is uniquely determined based on a probability vector $\vct{p}$ that associates each tuple a probability
|
||||
% variables are independent of each other (or disjoint if they are from the same block) and each variable $X$ is associated with a probability $\vct{p}(X) = \pd[X = 1]$. Thus, we are dealing with polynomials $\poly(\vct{X})$ that are annotations of a tuple in the result of a query $\query$ over a BIDB $\pxdb$ where $\vct{X}$ is the set of variables that occur in annotations of tuples of $\pxdb$.
|
||||
|
||||
|
|
@ -66,29 +71,30 @@ $\poly(\vct{X})$ = $\poly(X_{\block_1, 1},\ldots, X_{\block_1, \abs{\block_1}},$
|
|||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{definition}[Modding with a set]\label{def:mod-set}
|
||||
Let $S$ be a {\em set} of polynomials over $\vct{X}$. Then $\poly(\vct{X})\mod{S}$ is the polynomial obtained by taking the mod of $\poly(\vct{X})$ over {\em all} polynomials in $S$ (the order does not matter).
|
||||
Let $S$ be a {\em set} of polynomials over $\vct{X}$. Then $\poly(\vct{X})\mod{S}$ is the polynomial obtained by taking the mod of $\poly(\vct{X})$ over {\em all} polynomials in $S$ (order does not matter).
|
||||
\end{definition}
|
||||
For example when $S_0=\inset{X^2-X, Y^2-Y}$, taking the polynomial in~\cref{eq:poly-eg} mod $S_0$, we get $2X+3XY-2Y$.
|
||||
|
||||
%
|
||||
\begin{Definition}\label{def:mod-set-polys}
|
||||
Given the set of BIDB variables $\inset{X_{b,i}}$, define
|
||||
\[\mathcal{B}=\inset{X_{b,i}\cdot X_{b,j}|\text{ for every block } b \text{and } i\ne j},\]
|
||||
\[\mathcal{T}=\inset{X_{b,i}^2-X_{b,i}|\text{ for every block } b \text{and } i}.\]
|
||||
\[\mathcal{B}=\comprehension{X_{b,i}\cdot X_{b,j}}{\text{ for every block } b \text{and } i\ne j}\]
|
||||
\[\mathcal{T}=\comprehension{X_{b,i}^2-X_{b,i}}{\text{ for every block } b \text{and } i}\]
|
||||
\end{Definition}
|
||||
|
||||
%
|
||||
\begin{Definition}[Reduced \bi Polynomials]\label{def:reduced-bi-poly}
|
||||
Let $\poly(\vct{X})$ be a \bi-lineage polynomial.
|
||||
The reduced form $\rpoly(\vct{X})$ of $\poly(\vct{X})$ is defined as
|
||||
The reduced form $\rpoly(\vct{X})$ of $\poly(\vct{X})$ is:
|
||||
\begin{equation*}
|
||||
\rpoly(\vct{X}) = \smbOf{\poly(\vct{X})} \mod \mathcal{T} \mod \mathcal{B}%X_i^2 - X_i \mod X_{\block_s, t}X_{\block_s, u}
|
||||
\end{equation*}
|
||||
%for all $i$ in $[\numvar]$ and for all $s$ in $\ell$, such that for all $t, u$ in $[\abs{\block_s}]$, $t \neq u$.
|
||||
\end{Definition}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
Intuitively, in the reduced form all exponents $e > 1$ are reduced to $e = 1$ and, all monomials containing more than one variable from the same block $\block$ are dropped (tuples from the same block are disjoint events in \bis and, thus, any world containing more than one tuple from a block has $0$ probability and can be ignored). Note that for the special case of \tis, the second step (dropping monomials with variables from the same block) is not necessary since every block contains a single tuple.
|
||||
%
|
||||
Intuitively, in the reduced form, all exponents $e > 1$ are reduced to $e = 1$ and all monomials with multile variables from the same block $\block$ are dropped (any world containing more than one tuple from a block has $0$ probability and can be ignored).
|
||||
For the special case of \tis, the second step is not necessary since every block contains a single tuple.
|
||||
Alternatively, one can think of $\rpoly$ as the \abbrSMB of $\poly(\vct{X})$ when the product operator is idempotent.
|
||||
|
||||
%
|
||||
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% \begin{Definition}[$\rpoly(\vct{X})$] \label{def:qtilde}
|
||||
% Define $\rpoly(X_1,\ldots, X_\numvar)$ as the reduced version of $\poly(X_1,\ldots, X_\numvar)$, of the form
|
||||
|
|
@ -97,10 +103,10 @@ Alternatively, one can think of $\rpoly$ as the \abbrSMB of $\poly(\vct{X})$ whe
|
|||
% \[\poly(X_1,\ldots, X_\numvar) \mod X_1^2-X_1\cdots\mod X_\numvar^2 - X_\numvar.\]
|
||||
% \end{Definition}
|
||||
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{Example}\label{example:qtilde}
|
||||
Consider $\poly(X, Y) = (X + Y)(X + Y)$ where $X$ and $Y$ are from different blocks. Then the expanded derivation for $\rpoly(X, Y)$ is
|
||||
Consider $\poly(X, Y) = (X + Y)(X + Y)$ where $X$ and $Y$ are from different blocks. The expanded derivation for $\rpoly(X, Y)$ is
|
||||
\begin{align*}
|
||||
(&X^2 + 2XY + Y^2 \mod X^2 - X) \mod Y^2 - Y\\
|
||||
= ~&X + 2XY + Y^2 \mod Y^2 - Y\\
|
||||
|
|
@ -108,17 +114,19 @@ Consider $\poly(X, Y) = (X + Y)(X + Y)$ where $X$ and $Y$ are from different blo
|
|||
\end{align*}
|
||||
\end{Example}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%
|
||||
% Intuitively, $\rpoly(\textbf{X})$ is the \abbrSMB form of $\poly(\textbf{X})$ such that if any $X_j$ term has an exponent $e > 1$, it is reduced to $1$, i.e. $X_j^e\mapsto X_j$ for any $e > 1$.
|
||||
|
||||
|
||||
%
|
||||
%When considering $\bi$ input, it becomes necessary to redefine $\rpoly(\vct{X})$.
|
||||
|
||||
The usefulness of this will reduction become clear in \Cref{lem:exp-poly-rpoly}.
|
||||
|
||||
%
|
||||
\noindent The usefulness of this will reduction become clear in \Cref{lem:exp-poly-rpoly}.
|
||||
%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{Lemma}\label{lem:pre-poly-rpoly}
|
||||
When $\poly(X_1,\ldots, X_\numvar) = \sum\limits_{\vct{d} \in \{0,\ldots, B\}^\numvar}q_{\vct{d}} \cdot \prod\limits_{\substack{i = 1\\s.t. d_i\geq 1}}^{\numvar}X_i^{d_i}$, we have then that $\rpoly(X_1,\ldots, X_\numvar) = \sum\limits_{\vct{d} \in \{0,\ldots, B\}^\numvar} q_{\vct{d}}\cdot\prod\limits_{\substack{i = 1\\s.t. d_i\geq 1}}^{\numvar}X_i$.
|
||||
If
|
||||
$\poly(X_1,\ldots, X_\numvar) = \sum\limits_{\vct{d} \in \{0,\ldots, B\}^\numvar}q_{\vct{d}} \cdot \prod\limits_{\substack{i = 1\\s.t. d_i\geq 1}}^{\numvar}X_i^{d_i}$
|
||||
then
|
||||
$\rpoly(X_1,\ldots, X_\numvar) = \sum\limits_{\vct{d} \in \eta} q_{\vct{d}}\cdot\prod\limits_{\substack{i = 1\\s.t. d_i\geq 1}}^{\numvar}X_i$ \;\;\; for some $\eta \subseteq \{0,\ldots, B\}^\numvar$
|
||||
\end{Lemma}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
|
@ -127,13 +135,13 @@ Follows by the construction of $\rpoly$ in \cref{def:reduced-bi-poly}. \qed
|
|||
\end{proof}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
Note the following fact:
|
||||
\noindent Note the following fact:
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{Proposition}\label{proposition:q-qtilde} For any \bi-lineage polynomial $\poly(X_1, \ldots, X_\numvar)$ and all $\vct{w} \in \{0,1\}^\numvar$,
|
||||
\[
|
||||
\begin{Proposition}\label{proposition:q-qtilde} For any \bi-lineage polynomial $\poly(X_1, \ldots, X_\numvar)$ and all $\vct{w} \in \{0,1\}^\numvar$, it holds that
|
||||
$% \[
|
||||
\poly(\vct{w}) = \rpoly(\vct{w}).
|
||||
\]
|
||||
$% \]
|
||||
\end{Proposition}
|
||||
|
||||
|
||||
|
|
@ -170,7 +178,7 @@ to the variables $\vct{X}$. Intuitively, \Cref{lem:exp-poly-rpoly} states that w
|
|||
\begin{Corollary}\label{cor:expct-sop}
|
||||
If $\poly$ is a \bi-lineage polynomial, then the expectation of $\poly$, i.e., $\expct\pbox{\poly} = \rpoly\left(\prob_1,\ldots, \prob_\numvar\right)$ can be computed in $O(|\smbOf{\poly}|)$, where $|\poly|$ denotes the total number of multiplication/addition operators in $\poly$.
|
||||
\end{Corollary}
|
||||
\AH{What if $\poly$ is not in \abbrSMB form?}
|
||||
%\AH{What if $\poly$ is not in \abbrSMB form?}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
|
|
|||
|
|
@ -13,19 +13,19 @@ Denote the schema of $\db$ as $\sch(\db)$. A \textit{probabilistic database} $\p
|
|||
For a probabilistic database $\pdb = (\idb, \pd)$, the result of a query is the pair $(\query(\idb), \pd')$ where $\pd'$ is a probability distribution over $\query(\idb)$ that assigns to each possible query result the sum of the probabilities of the worlds that produce this answer:
|
||||
\[\forall \db \in \query(\idb): \pd'(\db) = \sum_{\db' \in \idb: \query(\db') = \db} \pd(\db') \]
|
||||
|
||||
Note that in this work we consider multisets, i.e., each possible world is a set of multiset relations and queries are evaluated using bag semantics. We will use K-relations to model multisets. A \emph{K-relation}~\cite{DBLP:conf/pods/GreenKT07} is a relation whose tuples are annotated with elements from a commutative semiring $\semK = (\domK, \addK, \multK, \zeroK, \oneK)$. A commutative semiring is a structure with a domain $\domK$ and associative and commutative binary operations $\addK$ and $\multK$ such that $\multK$ distributes over $\addK$, $\zeroK$ is the identity of $\addK$, $\oneK$ is the identity of $\multK$, and $\zeroK$ annihilates all elements of $\domK$ when being combined with $\multK$.
|
||||
Note that in this work we consider multisets, i.e., each possible world is a set of multiset relations and queries are evaluated using bag semantics. We will use K-relations to model multisets. A \emph{K-relation}~\cite{DBLP:conf/pods/GreenKT07} is a relation whose tuples are annotated with elements from a commutative semiring $\semK = (\domK, \addK, \multK, \zeroK, \oneK)$. A commutative semiring is a structure with a domain $\domK$ and associative and commutative binary operations $\addK$ and $\multK$ such that $\multK$ distributes over $\addK$, $\zeroK$ is the identity of $\addK$, $\oneK$ is the identity of $\multK$, and $\zeroK$ annihilates all elements of $\domK$ when combined by $\multK$.
|
||||
Let $\udom$ be a countable domain of values.
|
||||
Formally, an n-ary $\semK$-relation over $\udom$ is a function $\rel: \udom^n \to \domK$ with finite support $\support{\rel} = \{ \tup \mid \rel(\tup) \neq \zeroK \}$.
|
||||
A $\semK$-database is a set of $\semK$-relations. It will be convenient to also interpret a $\semK$-database as a function from tuples to annotations. Thus, $\rel(t)$ ($\db(t)$) denotes the annotation associated by $\semK$-relation $\rel$ ($\semK$-database $\db$) to tuple $t$.
|
||||
We review the semantics of positive relational algebra queries over $\semK$-relations below.
|
||||
A $\semK$-database is a set of $\semK$-relations. It will be convenient to also interpret a $\semK$-database as a function from tuples to annotations. Thus, $\rel(t)$ (resp., $\db(t)$) denotes the annotation associated by $\semK$-relation $\rel$ ($\semK$-database $\db$) to $t$.
|
||||
We review positive relational algebra semantics for $\semK$-relations below.
|
||||
|
||||
|
||||
Consider the semiring $\semN = (\domN,+,\times,0,1)$ of natural numbers. $\semN$-databases are used to model bag semantics by annotating each tuple with its multiplicity. A probabilistic $\semN$-database ($\semN$-PDB) is a PDB where each possible world is an $\semN$-database. We will study the problem of evaluating statistical moments of query results over such databases. Specifically, given a probabilistic $\semN$-database $\pdb = (\idb, \pd)$, query $\query$, and possible result tuple $t$, we treat $\query(\db)(t)$ as a random $\semN$-valued variable and are interested in computing its expectation $\expct_{\idb \sim \pd}[\query(\db)(t)]$:
|
||||
|
||||
Consider the semiring $\semN = (\domN,+,\times,0,1)$ of natural numbers. $\semN$-databases model bag semantics by annotating each tuple with its multiplicity. A probabilistic $\semN$-database ($\semN$-PDB) is a PDB where each possible world is an $\semN$-database. We study the problem of computing statistical moments for query results over such databases. Specifically, given a probabilistic $\semN$-database $\pdb = (\idb, \pd)$, query $\query$, and possible result $t$, we treat $\query(\db)(t)$ as a random $\semN$-valued variable and are interested in computing its expectation $\expct_{\idb \sim \pd}[\query(\db)(t)]$:
|
||||
%
|
||||
\begin{align}\label{eq:bag-expectation}
|
||||
\expct_{\idb \sim \pd}[\query(\db)(t)] = \sum_{\db \in \idb} \query(\db)(t) \cdot \pd(\db)
|
||||
\end{align}
|
||||
|
||||
%
|
||||
Intuitively, the expectation of $\query(\db)(t)$ is the number of duplicates of $t$ we expect to find in result of query $\query$.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
|
@ -34,7 +34,7 @@ Intuitively, the expectation of $\query(\db)(t)$ is the number of duplicates of
|
|||
\subsubsection{$\semK$-relational Query Semantics}
|
||||
For completeness, we briefly review the semantics for $\raPlus$ queries over $\semK$-relations~\cite{DBLP:conf/pods/GreenKT07}.
|
||||
We use $\evald{\cdot}{\db}$ to denote the result of evaluating query $\query$ over $\semK$-database $\db$. In the definition shown below, we assume that tuples are of appropriate arity and use $\project_A(\tup)$ to denote the projection of tuple $\tup$ on a list of attributes $A$. Furthermore, $\theta(\tup)$ denotes the (boolean) result of evaluating condition $\theta$ over $\tup$.
|
||||
|
||||
%
|
||||
\begin{align*}
|
||||
& \evald{\project_A(\rel)}{\db}(\tup) & & = & & \sum_{\tup': \project_A(\tup') = \tup} \evald{\rel}{\db}(\tup') \\
|
||||
& \evald{(\rel_1 \union \rel_2)}{\db}(\tup) & & = & & \evald{\rel_1}{\db}(\tup) \addK \evald{\rel_2}{\db}(\tup) \\
|
||||
|
|
@ -50,16 +50,22 @@ We use $\evald{\cdot}{\db}$ to denote the result of evaluating query $\query$ ov
|
|||
\subsubsection{$\semNX$ as a Representation System}\label{sec:semnx-as-repr}
|
||||
|
||||
Let $\semNX$ denote the set of polynomials over variables $\vct{X}$ with natural number coefficients and exponents.
|
||||
Consider now the semiring $(\semNX, +, \cdot, 0, 1)$ whose domain is $\semNX$ and addition and multiplication are standard addition and multiplication of polynomials. We will utilize $\semNX$-PDB $\pxdb$, defined as the tuple $(\db, \pd)$, where $\semNX$-database $\db$ is paired with probability distribution $\pd$.
|
||||
For supplementary material leading up to~\Cref{prop:expection-of-polynom}, see~\Cref{subsec:supp-mat-background}.
|
||||
Consider now the semiring $(\semNX, +, \cdot, 0, 1)$ whose domain is $\semNX$ and with the standard addition and multiplication of polynomials.
|
||||
We will utilize $\semNX$-PDB $\pxdb$, defined as the tuple $(\db, \pd)$, where $\semNX$-database $\db$ is paired with probability distribution $\pd$.
|
||||
We denote by $\polyForTuple$ the annotation of tuple $t$ in the result of $\query$ (i.e., $\polyForTuple = \query(\pxdb)(t)$) and as before, interpret it as a function $\polyForTuple: \{0,1\}^{|\vct X|} \rightarrow \semN$ from vectors of variable assignments to the corresponding value of the annotating polynomial.
|
||||
$\semNX$-PDBs, a function $\rmod$, which takes an $\semNX$-PDB input and outputs an equivalent $\semN$-PDB are formally defined in \Cref{subsec:supp-mat-background}.
|
||||
|
||||
Assume a function $\rmod$, which takes an $\semNX$-PDB input and outputs an equivalent $\semN$-PDB.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{Proposition}[Expectation of polynomials]\label{prop:expection-of-polynom}
|
||||
Given an $\semN$-PDB $\pdb = (\idb,\pd)$ and $\semNX$-PDB $\pxdb = (\db,\pd')$ such that $\rmod(\pxdb) = \pdb$, we have:
|
||||
\[ \expct_{\idb \sim \pd}[\query(\db)(t)] = \expct_{\vct{W} \sim \pd'}\pbox{\poly(\vct{W})} \]
|
||||
Given an $\semN$-PDB $\pdb = (\idb,\pd)$ and $\semNX$-PDB $\pxdb = (\db,\pd')$ where $\rmod(\pxdb) = \pdb$:
|
||||
\[ \expct_{\idb \sim \pd}[\query(\db)(t)] = \expct_{\vct{w} \sim \pd'}\pbox{\polyForTuple(\vct{w})} \]
|
||||
\end{Proposition}
|
||||
\noindent A formal proof of \Cref{prop:expection-of-polynom} is given in \Cref{subsec:expectation-of-polynom-proof}.
|
||||
This proposition shows that computing the expected multiplicity of a query result tuple is equivalent to computing the expectation of a polynomial (for that tuple) from a probability distribution over all possible assignments of variables in the polynomial to $\{0,1\}$.
|
||||
We focus on this problem exclusively from now on, assume an implicit result tuple, and drop the subscript from $\polyForTuple$ (i.e., $\poly$ is used as a polynomial from this point on).
|
||||
|
||||
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
|
@ -69,14 +75,14 @@ Assume a function $\rmod$, which takes an $\semNX$-PDB input and outputs an equi
|
|||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{Definition}[\tis and \bis]\label{def:tidbs-and-bidbs}
|
||||
A \emph{\ti} $\pxdb = (\db, \pd)$ is an $\semNX$-PDB such that (i) every tuple is annotated with either $0$ or a unique variable $X_i$ and (ii) the probability distribution $\pd$ is such that all variables are independent.
|
||||
|
||||
A \emph{\bi} $\pxdb = (\db, \pd)$ is an $\semNX$-PDB such that (i) every tuple is annotated with either $0$ or a unique variable $X_i$ and (ii) that the tuples $\tup$ of $\pxdb$ for which $\pxdb(\tup) \neq 0$ can be partitioned into a set of blocks such that variables from separate blocks are independent of each other and variables from the same blocks are disjoint events.
|
||||
\BG{Should this be written in math?}
|
||||
\end{Definition}
|
||||
\subsubsection{\tis and \bis}
|
||||
In this paper, we focus on two popular forms of PDB: Block-Independent (\bi) and Tuple-Independent (\ti) PDBs.
|
||||
%
|
||||
A \bi $\pxdb = (\db, \pd)$ is an $\semNX$-PDB such that (i) every tuple is annotated with either $0$ or a unique variable $X_i$ and (ii) that the tuples $\tup$ of $\pxdb$ for which $\pxdb(\tup) \neq 0$ can be partitioned into a set of blocks such that variables from separate blocks are independent of each other and variables from the same blocks are disjoint events.
|
||||
%
|
||||
A \emph{\ti} is a \bi where each block contains exactly one tuple.
|
||||
\Cref{subsec:supp-mat-ti-bi-def} explains \tis and \bis in greater detail.
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
For supplementary material on~\Cref{def:tidbs-and-bidbs}, see~\Cref{subsec:supp-mat-ti-bi-def}.
|
||||
|
||||
|
||||
\input{poly-form.tex}
|
||||
|
|
@ -84,7 +90,7 @@ For supplementary material on~\Cref{def:tidbs-and-bidbs}, see~\Cref{subsec:supp-
|
|||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{Expression Trees}\label{sec:expression-trees}
|
||||
|
||||
In the following we will make use of expression trees to encode polynomials which we define formally in this subsection. Later on in~\Cref{sec:gen} we discuss using circuits.
|
||||
In this section, we formally define expression trees, an encoding of polynomials that we use throughout much of the paper before generalizing to circuits in~\Cref{sec:gen}.
|
||||
For illustrative purposes consider the polynomial $\poly(\vct{X}) = 2X_1^2 + 3X_1X_2 - 2X_2^2$ over $\vct{X} = [X_1, X_2]$.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
|
@ -95,19 +101,18 @@ tree, whose internal nodes are from the set $\{+, \times\}$, with leaf nodes bei
|
|||
\end{Definition}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
We ignore the remaining fields (\vari{partial} and \vari{weight}) for now. Their purpose will become clear in~\Cref{sec:algo}. Note that $\etree$ need not encode an expression in standard monomial basis. For instance, $\etree$ could represent a compressed form of the running example, such as $(X_1 + 2X_2)(2X_1 - X_2)$.
|
||||
We ignore the remaining fields (\vari{partial} and \vari{weight}) until \Cref{sec:algo}. Note that $\etree$ need not encode an expression in standard monomial basis. For instance, $\etree$ could represent a compressed form of the running example, such as $(X_1 + 2X_2)(2X_1 - X_2)$.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{Definition}[poly$(\cdot)$]\label{def:poly-func}
|
||||
Denote $poly(\etree)$ to be the function that takes as input expression tree $\etree$ and outputs its corresponding polynomial in \abbrSMB. $poly(\cdot)$ is recursively defined on $\etree$ as follows, where $\etree_\lchild$ and $\etree_\rchild$ denote the left and right child of $\etree$ respectively.
|
||||
|
||||
%
|
||||
% \begin{align*}
|
||||
% &\etree.\type = +\mapsto&& \polyf(\etree_\lchild) + \polyf(\etree_\rchild)\\
|
||||
% &\etree.\type = \times\mapsto&& \polyf(\etree_\lchild) \cdot \polyf(\etree_\rchild)\\
|
||||
% &\etree.\type = \var \text{ OR } \tnum\mapsto&& \etree.\val
|
||||
% \end{align*}
|
||||
|
||||
|
||||
%
|
||||
\begin{equation*}
|
||||
\polyf(\etree) = \begin{cases}
|
||||
\polyf(\etree_\lchild) + \polyf(\etree_\rchild) &\text{ if \etree.\type } = +\\
|
||||
|
|
@ -131,18 +136,16 @@ For our running example, $\etreeset{\smb} = \{2X_1^2 + 3X_1X_2 - 2X_2^2, (X_1 +
|
|||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{Problem Definition}\label{sec:problem-definition}
|
||||
|
||||
We are now ready to formally state the main problem addressed in this work.
|
||||
We are now ready to formally state our main problem.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{Definition}[The Expected Result Multiplicity Problem]\label{def:the-expected-multipl}
|
||||
Let $\vct{X} = (X_1, \ldots, X_n)$, and $\pdb$ be an $\semNX$-PDB over $\vct{X}$ with probability distribution $\pd$ over assignments $\vct{X} \to [0,1]$, $\query$ an n-ary query, and $t$ an n-ary tuple.
|
||||
The \expectProblem is defined as follows:
|
||||
\AH{I think we mean $\poly(\vct{X}) = \query(\pxdb)(t)$ instead of $\poly(\vct{X}) = \query(\pdb)(t)$. I changed the following to reflect this.}
|
||||
\BG{Correct}
|
||||
\begin{itemize}
|
||||
\item \textbf{Input}: Given an expression tree $\etree \in \etreeset{\smb}$ for $\poly(\vct{X}) = \query(\pxdb)(t)$
|
||||
\item \textbf{Output}: $\expct_{\vct{X} \sim \pd}[\poly(\vct{X})]$
|
||||
\end{itemize}
|
||||
% \AH{I think we mean $\poly(\vct{X}) = \query(\pxdb)(t)$ instead of $\poly(\vct{X}) = \query(\pdb)(t)$. I changed the following to reflect this.}
|
||||
% \BG{Correct}
|
||||
\\\hspace*{5mm}\textbf{Input}: An expression tree $\etree \in \etreeset{\smb}$ for $\poly(\vct{X}) = \query(\pxdb)(t)$
|
||||
\\\hspace*{5mm}\textbf{Output}: $\expct_{\vct{X} \sim \pd}[\poly(\vct{X})]$
|
||||
\end{Definition}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
|
|
|
|||
27
single_p.tex
27
single_p.tex
|
|
@ -1,5 +1,5 @@
|
|||
|
||||
%root: main.tex
|
||||
%!TEX root=./main.tex
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{Single $\prob$ value}
|
||||
|
|
@ -15,7 +15,9 @@ Fix $p\in (0,1)$. Then assuming \Cref{conj:graph} is true, then any algorithm th
|
|||
%\begin{proof}[Proof of Corollary ~\ref{th:single-p-gen-k}]
|
||||
%Consider $\poly^3_{G}$ and $\poly' = 1$ such that $\poly'' = \poly^3_{G} \cdot \poly'$. By \Cref{th:single-p}, query $\poly''$ with $\kElem = 4$ has $\Omega(\numvar^{\frac{4}{3}})$ complexity.
|
||||
%\end{proof}
|
||||
The above shows the hardness for a very specific query polynomial but it is easy to come up with an infinite family of hard query polynomials by `embedding' $\rpoly_{G}^3$ into an infinite family of trivial query polynomials. However, unlike \Cref{thm:mult-p-hard-result} the above result does not show that computing $\rpoly_{G}^3(\prob,\dots,\prob)$ for a fixed $p\in (0,1)$ is \sharpwonehard. By contrast, in \Cref{sec:algo} we show that if we are willing to compute an approximation that this problem (and indeed solving our problem for a much more general setting) is in linear time.
|
||||
The above shows the hardness for a very specific query polynomial but it is easy to come up with an infinite family of hard query polynomials by `embedding' $\rpoly_{G}^3$ into an infinite family of trivial query polynomials.
|
||||
Unlike \Cref{thm:mult-p-hard-result} the above result does not show that computing $\rpoly_{G}^3(\prob,\dots,\prob)$ for a fixed $p\in (0,1)$ is \sharpwonehard.
|
||||
However, in \Cref{sec:algo} we show that if we are willing to compute an approximation that this problem (and indeed solving our problem for a much more general setting) is in linear time.
|
||||
|
||||
%\AH{@atri needs to put in the result for triangles of $\numvar^{\frac{4}{3}}$ runtime.}
|
||||
We will prove the above result by the following reduction:
|
||||
|
|
@ -27,19 +29,20 @@ in $O\inparen{T(\numedge) + \numedge}$ time.
|
|||
\end{Theorem}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{proof}[Proof of \Cref{th:single-p}]
|
||||
For the sake of contradiction, let us assume that for any $G$, we can compute $\rpoly_{G}^3(\prob,\dots,\prob)$ in $o\inparen{m^{1+\eps_0}}$ time.
|
||||
\begin{proof}[Proof of \Cref{th:single-p-hard}]
|
||||
For the sake of contradiction, assume that for any $G$, we can compute $\rpoly_{G}^3(\prob,\dots,\prob)$ in $o\inparen{m^{1+\eps_0}}$ time.
|
||||
Let $G$ be the input graph. It is easy to see that one can compute the expression tree for $\poly_{G}^3(\vct{X})$ in $O(m)$ time. Then by \Cref{th:single-p} we can compute $\numocc{G}{\tri}$, $\numocc{G}{\threepath}$ and $\numocc{G}{\threedis}$ in further time $o\inparen{m^{1+\eps_0}}+O(m)$. Thus, the overall, reduction takes $o\inparen{m^{1+\eps_0}}+O(m)= o\inparen{m^{1+\eps_0}}$ time, which violates \Cref{conj:graph}.
|
||||
\end{proof}
|
||||
\qed
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
Before moving on to prove \Cref{th:single-p-hard}, let us state the results, lemmas and defintions that will be useful in the proof.
|
||||
In other words, if \Cref{th:single-p} holds, then so must \Cref{th:single-p-hard}.
|
||||
Before we move on to the proof itself, we state the results, lemmas, and defintions that will be useful in the proof.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsubsection{Preliminaries and Notation}
|
||||
|
||||
We need to list all possible edge patterns in an arbitrary $G$ consisting of at most three distinct edges. We have already seen $\tri,\threepath$ and $\threedis$, so here we define the remaining patterns:
|
||||
We need all the possible edge patterns in an arbitrary $G$ with at most three distinct edges. We have already seen $\tri,\threepath$ and $\threedis$, so we define the remaining patterns:
|
||||
|
||||
\begin{itemize}
|
||||
\item Single Edge $\left(\ed\right)$
|
||||
|
|
@ -47,7 +50,7 @@ We need to list all possible edge patterns in an arbitrary $G$ consisting of at
|
|||
\item 2-matching ($\twodis$)
|
||||
%\item Triangle ($\tri$)
|
||||
%\item 3-path ($\threepath$)
|
||||
\item 3-star ($\oneint$)--this is the graph that results when all three edges share exactly one common endpoint. The remaining endpoint for each edge is disconnected from any endpoint of the three edges.
|
||||
\item 3-star ($\oneint$)--this is the graph that results when all three edges share exactly one common endpoint. The remaining endpoint for each edge is disconnected from any endpoint of the remaining two edges.
|
||||
\item Disjoint Two-Path ($\twopathdis$)--this subgraph consists of a two path and a remaining disjoint edge.
|
||||
%\item 3-matching ($\threedis$)--this subgraph is composed of three disjoint edges.
|
||||
\end{itemize}
|
||||
|
|
@ -73,7 +76,7 @@ For any graph $G$, the following formulas for $\numocc{G}{H}$ for their respecti
|
|||
|
||||
\subsubsection{The proofs}
|
||||
|
||||
Note that $\rpoly_{G}^3(\prob,\ldots, \prob)$ as a polynomial in $\prob$ has degree at most six. Next, we figure out the exact coefficients since this would be useful in our arguments:
|
||||
Note that $\rpoly_{G}^3(\prob,\ldots, \prob)$ as a polynomial in $\prob$ has degree at most six. Next, we figure out the exact coefficients (i.e., $c_i$) since this would be useful in our arguments:
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{Lemma}\label{lem:qE3-exp}
|
||||
%When we expand $\poly_{G}^3(\vct{X})$ out and assign all exponents $e \geq 1$ a value of $1$, we have the following result,
|
||||
|
|
@ -88,12 +91,12 @@ For any $p$, we have:
|
|||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{proof}%[Proof of \Cref{lem:qE3-exp}]
|
||||
By definition we have that
|
||||
\[\poly_{G}^3(\vct{X}) = \sum_{\substack{(i_1, j_1),\\ (i_2, j_2),\\ (i_3, j_3) \in E}} \prod_{\ell = 1}^{3}X_{i_\ell}X_{j_\ell}.\]
|
||||
Hence $\rpoly_{G}^3(\vct{X})$ has degree six. Note that the monomial $\prod_{\ell = 1}^{3}X_{i_\ell}X_{j_\ell}$ will contribute to the coefficient of $p^i$ in $\rpoly_{G}^3(\vct{X})$, where $i$ is the number of distinct variables in the monomial.
|
||||
\[\poly_{G}^3(\vct{X}) = \sum_{\substack{(i_1, j_1), (i_2, j_2), (i_3, j_3) \in E}}~\; \prod_{\ell = 1}^{3}X_{i_\ell}X_{j_\ell}.\]
|
||||
Hence $\rpoly_{G}^3(\vct{X})$ has degree six. Note that the monomial $\prod_{\ell = 1}^{3}X_{i_\ell}X_{j_\ell}$ will contribute to the coefficient of $p^\nu$ in $\rpoly_{G}^3(\vct{X})$, where $\nu$ is the number of distinct variables in the monomial.
|
||||
%Rather than list all the expressions in full detail, let us make some observations regarding the sum.
|
||||
Let $e_1 = (i_1, j_1), e_2 = (i_2, j_2), e_3 = (i_3, j_3)$. Notice that each expression in the sum consists of a triple $(e_1, e_2, e_3)$. There are three forms the triple $(e_1, e_2, e_3)$ can take (and in each case, we will account for their contribution to $\rpoly_{G}^3(\vct{X})$).
|
||||
Let $e_1 = (i_1, j_1), e_2 = (i_2, j_2), e_3 = (i_3, j_3)$. Notice that each expression in the sum is a triple $(e_1, e_2, e_3)$. There are three forms the triple $(e_1, e_2, e_3)$ can take (and in each case, we account for their contribution to $\rpoly_{G}^3(\vct{X})$).
|
||||
|
||||
\textsc{case 1:} $e_1 = e_2 = e_3$, where all edges are the same. There are exactly $\numedge=\numocc{G}{\ed}$ such triples, each with a $\prob^2$ factor in $\rpoly_{G}^3\left(\prob,\ldots, \prob\right)$.
|
||||
\textsc{case 1:} $e_1 = e_2 = e_3$ (all edges are the same). There are exactly $\numedge=\numocc{G}{\ed}$ such triples, each with a $\prob^2$ factor in $\rpoly_{G}^3\left(\prob,\ldots, \prob\right)$.
|
||||
|
||||
\textsc{case 2:} This case occurs when there are two distinct edges of the three, call them $e$ and $e'$. When there are two distinct edges, there is then the occurence when $2$ variables in the triple $(e_1, e_2, e_3)$ are bound to $e$. There are three combinations for this occurrence in $\poly_{G}^3(\vct{X})$. Analogusly, there are three such occurrences in $\poly_{G}^3(\vct{X})$ when there is only one occurrence of $e$, i.e. $2$ of the variables in $(e_1, e_2, e_3)$ are $e'$. %Again, there are three combinations for this.
|
||||
This implies that all $3 + 3 = 6$ combinations of two distinct edges $e$ and $e'$ contribute to the same monomial in $\rpoly_{G}^3$. % consist of the same monomial in $\rpoly$, i.e. $(e_1, e_1, e_2)$ is the same as $(e_2, e_1, e_2)$.
|
||||
|
|
|
|||
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Reference in New Issue